Z-domain analysis of PLL

Hi,


For the s-domain analysis, the low pass filter has a minimal state space
realization:

dxf/dt = Af * xf +Bf *uf
yf = Cf *xf

The linear system G(s) is represented by
dx/dt= A * x +B *u
y = C *x

I would like to ask about how to transform the PLL configuration into the
Lur'e system, and then find the corresponding G(z)? From there,how to find
the state-space matrices A,B,C and D which are in terms of the filter's
parameters (Af,Bf,Cf,Df) and the VCO's parameters (Av,Bv,Cv,Dv)? G(z) is a
linear system which consists of loop filter and VCO.

Thanks.


	 

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On Wed, 11 Feb 2015 07:21:49 -0600, algae11 wrote:

> Hi,
> 
> 
> For the s-domain analysis, the low pass filter has a minimal state space
> realization:
> 
> dxf/dt = Af * xf +Bf *uf yf = Cf *xf
> 
> The linear system G(s) is represented by dx/dt= A * x +B *u y = C *x
> 
> I would like to ask about how to transform the PLL configuration into
> the Lur'e system, and then find the corresponding G(z)? From there,how
> to find the state-space matrices A,B,C and D which are in terms of the
> filter's parameters (Af,Bf,Cf,Df) and the VCO's parameters
> (Av,Bv,Cv,Dv)? G(z) is a linear system which consists of loop filter and
> VCO.
> 
> Thanks.

I'm not familiar with the term "Lur'e system".  Nor am I sure exactly what 
you want to do or what you're trying to achieve.

If you already have the system realized in continuous-time, then it's 
probably a good idea to keep it there (while keeping in mind that a PLL is 
a time-varying, nonlinear system, so you _know_ that you're dealing with 
an approximation). 

If you are planning on implementing the controller portion (loop filter 
and any associated intentional nonlinearities, like assisted locking or 
whatever) in sampled time, then you don't want to take a continuous-time 
model of a loop filter that you may have designed for some other PLL -- 
you want to model your VCO (or NCO, or whatever) and your phase detector 
in sampled time, and then design your loop filter in the z domain.

So -- tell us more, and maybe we can tell you more.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com